Finite Group Databasebeta

← #30601963

Subgroups — #30601963

Order 15616.

510 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462008C212yesmaximal
2#21461695C211yesmaximal, characteristic
3#21461555C211yesmaximal, characteristic
4#21461259C212yesmaximal
5C2×#16840000C211yesmaximal, characteristic
6C2×#14519845—21no—
7D61⋊(C22⋊C8)C2211yescharacteristic
8C2×#14519831—21no—
9#16840704C2212yes—
10C4×C22⋊#6923C2211yescharacteristic
11#16840640—24no—
12#16840636—21no—
13(C2×#6923)⋊C8—21no—
14#16840616—24no—
15C22×C8×D61C2211yescharacteristic
16C2×#14519753C2212yes—
17#16840000—21no—
18#16840000—21no—
19#16840000—21no—
20#16840000—21no—
21C2×#14519711C411yescharacteristic, complemented
22C2×#14519701C412yescomplemented
23C22×Q16×C61C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×C4⋊#6923—22no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923D412yes—
28C2×C4×#6923—21no—
29#14519845—42no—
30#14519834D412yes—
31C22⋊#92420—21no—
32#14519831—42no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—21no—
35C22×C4×D61C2311yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C2×C8×D61—42no—
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61D411yescharacteristic
42#14519753—22no—
43#14519753—22no—
44#14519753—22no—
45#14519753—22no—
46C2×C4⋊Dic61C2×C412yes—
47C2×#11758196—22no—
48C2×#11758198—22no—
49#14519711—21no—
50#14519711—21no—
51#14519701—22no—
52#14519701—22no—
53C22×Q8×C61C2×C412yes—
54C2×Q16×C61—21no—
55C2×Q16×C61—21no—
56C2×Q16×C61—42no—
57C22×C8×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—22no—
60C22×#6923—42no—
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—42no—
64C2×Q8×C61—42no—
65C22×C4×C61—22no—
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61#4111yescharacteristic
69C2×C4×D61—41no—
70C2×C4×D61—41no—
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×D411yescharacteristic
73C2×C4×D61C2×D411yescharacteristic
74C2×C4×D61#4111yescharacteristic
75C22⋊#6923—24no—
76C22⋊#6923—44no—
77C2×#92420—22no—
78C4⋊#6923—42no—
79C4⋊#6923—42no—
80C4×#6923—41no—
81C4×#6923—41no—
82D61⋊C8—41no—
83D61⋊C8—41no—
84Q16×C61—42no—
85Q16×C61—42no—
86C2×C8×C61—22no—
87C2×C8×C61C22⋊C411yescharacteristic
88C2×C8×C61C22⋊C411yescharacteristic
89#11758198—42no—
90#11758198—42no—
91C4⋊Dic61—22no—
92C4⋊Dic61—22no—
93#11758196—42no—
94#11758196—42no—
95C2×#92418—41no—
96C2×#92418C22⋊C411yescharacteristic
97C2×#92418C22⋊C411yescharacteristic
98C8×D61—22no—
99C8×D61—22no—
100C8×D61—41no—
101C8×D61—41no—
102C23×C61C4×D411yescharacteristic
103C22×D61—41no—
104C22×D61—41no—
105C22×D61#12111yescharacteristic
106C22×D61#12711yescharacteristic
107C22×D61#13714yes—
108C2×#6923—42no—
109C2×#6923—42no—
110C2×#6923—42no—
111C2×#6923—82no—
112C2×#6923—82no—
113C2×#6923—82no—
114Q8×C61—42no—
115Q8×C61—42no—
116Q8×C61—82no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—42no—
120C2×C4×C61—82no—
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61C4×D411yescharacteristic
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—82no—
126C2×Dic61#12111yescharacteristic
127C2×Dic61#12411yescharacteristic
128C4×D61—21no—
129C4×D61—21no—
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—41no—
133C4×D61—41no—
134#92420—42no—
135C8×C61—21no—
136C8×C61#9912yes—
137#92418—21no—
138#92418—21no—
139#9784—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#39011yescharacteristic
143C22×C61#44112yes—
144C2×D61—21no—
145C2×D61—21no—
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—44no—
149C2×D61#47012yes—
150C2×D61#49712yes—
151#6923—82no—
152#6923—162no—
153C4×C61—21no—
154C4×C61—82no—
155C4×C61C2×#9911yescharacteristic
156C4×C61#41011yescharacteristic
157Dic61—21no—
158Dic61—21no—
159Dic61—82no—
160C2×#467—611no—
161#1996—612no—
162#1807—611no—
163#1752—611no—
164#1520—612no—
165C2×C61—22no—
166C2×C61#196411yescharacteristic
167C2×C61#197111yescharacteristic
168C2×C61#197611yescharacteristic
169D61—22no—
170D61—22no—
171C22×Q16#692311yescharacteristic, complemented
172C23×C8—611no—
173C2×C22⋊Q8—612no—
174#467—1221no—
175#467—1221no—
176#467—1221no—
177#467—1221no—
178C2×#107—611no—
179C2×#106—1221no—
180C2×#103—612no—
181C2×C22⋊C8—611no—
182C2×#97—1221no—
183#380—612no—
184C4×C22⋊C4—611no—
185#342—1224no—
186#340—1221no—
187#336—1221no—
188#328—1224no—
189C61#978411yescharacteristic, complemented, Sylow
190C22×Q8C2×#692312yes—
191C23×C4—611no—
192C2×Q16—21no—
193C2×Q16—21no—
194C2×Q16—42no—
195C22×C8—611no—
196C22×C8—611no—
197C22×C8—611no—
198C22×C8—611no—
199C22×C8—611no—
200C22×C8—1221no—
201C22×C8—2442no—
202C22×C8C2×#692311yescharacteristic
203C22⋊Q8—1222no—
204C22⋊Q8—1222no—
205C22⋊Q8—1222no—
206C22⋊Q8—1222no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—1222no—
209C2×C4⋊C4—1222no—
210C2×C22⋊C4—612no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C42—1221no—
214#107—1221no—
215#107—1221no—
216#106—2442no—
217#103—1222no—
218#103—1222no—
219C22⋊C8—612no—
220C22⋊C8—1221no—
221#97—2442no—
222#95—1221no—
223#95—1222no—
224C24—611no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—42no—
229C22×C4—22no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4—2441no—
238C22×C4—2441no—
239C22×C4—2442no—
240C22×C4C22×#692311yescharacteristic
241Q16—42no—
242Q16—42no—
243C2×C8—22no—
244C2×C8—611no—
245C2×C8—611no—
246C2×C8—1222no—
247C2×C8—1222no—
248C2×C8—1222no—
249C2×C8—2441no—
250C2×C8—2441no—
251C2×C8—2441no—
252C2×C8—2441no—
253C2×C8—2441no—
254C2×C8C22⋊#692311yescharacteristic
255C2×C8C22⋊#692311yescharacteristic
256C4⋊C4—1222no—
257C4⋊C4—1222no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2442no—
262C22⋊C4—1224no—
263C22⋊C4—2442no—
264C22⋊C4—2442no—
265C22⋊C4—2444no—
266C42—2441no—
267C42—2441no—
268C23—611no—
269C23—611no—
270C23—614no—
271C23—2441no—
272C23—2441no—
273C23D4×#692311yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—611no—
282C2×C4—611no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—4882no—
288C2×C4—4882no—
289C2×C4—4882no—
290C2×C4—4882no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4C2×C22⋊#692311yescharacteristic
299C2×C4D4×#692311yescharacteristic
300C8—21no—
301C8—1221no—
302C8—1221no—
303C8—2442no—
304C8#1451983512yes—
305C22—21no—
306C22—21no—
307C22—612no—
308C22—612no—
309C22—1221no—
310C22—1221no—
311C22—1221no—
312C22—1221no—
313C22—2444no—
314C22#1684071411yescharacteristic
315C22#1684080212yes—
316C4—21no—
317C4—82no—
318C4—1221no—
319C4—1221no—
320C4—9762no—
321C4—4882no—
322C4—4882no—
323C4C2×#1451983511yescharacteristic
324C4#1684074211yescharacteristic
325C2—22no—
326C2—1222no—
327C2—1222no—
328C2#2146195811yescharacteristic
329C2#2146196711yescharacteristic
330C2#2146197811yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×Q16×C61C4
socleC22×C61#390
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC22×Q16#6923
hypocenterC61#9784
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
21. C2×#14519711C4semidirect
22. C2×#14519701C4semidirect
23. C22×Q16×C61C4semidirect
171. C22×Q16#6923semidirect
189. C61#9784semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214620082
#214616952
#214615552
#214612592
C2×#168400002
#978461